Experimental study on failure characteristics and rheological properties of pillar-like rock samples with different shapes

The stability of coal pillar is extremely important to the control of rock strata movement and surface subsidence. It is of great significance for mining design to analyze the stability and failure characteristics of coal and rock pillars left after mining and to study the failure characteristics and rheological properties of coal and rock with different shapes. In this paper, based on uniaxial compression and rheological tests on rock samples, the rheological properties of rock samples with different shapes were discussed by using the nonlinear theoretical mechanics and damage theory, and the rheological mechanical characteristics of coarse yellow sandstone samples under the action of different free surface areas and the same loading contact area were investigated by means of experimental research, theoretical analysis and numerical simulation. The following conclusions were drawn: the failure characteristics and dynamic change process of rock samples with different shapes under the same loading contact area are obtained by uniaxial compression test and multi-stage rheological loading. The uniaxial compressive strengths of rock samples with the same loading contact surface area and different free surface areas are inversely proportional to their free surface areas. For the round sample, the stress level in the rheological test is obviously lower than the instantaneous peak uniaxial compression strength, while for the other samples, the stress level in the rheological test is close to the instantaneous peak uniaxial compression strength. For rock all these samples, both the ratio of steady-state rheological time to final failure time and the deformation degree decrease with the increase of free surface area.

As mining-induced environmental deterioration and geological disasters become increasingly severe, appropriate ways of reducing subsidence, controlling loss and realizing green mining have gained wide attention from researchers.The core of these problems is to control the movement of overburden left after, mining and reduce the damage to overburden and surface 1,2 .In order to prevent or reduce the related natural disasters caused by the rheological instability of coal and rock pillars of underground engineering such as surface subsidence, rock mass slide and ground instability, in-depth and long-lasting research has been conducted on the properties of rock mechanics and achieved many significant results in the aspects of strength, instability, deformation, damage, fracture and destruction.
Some researchers conducted laboratory tests to explore the mechanical characteristics of rock failure under true triaxial loading and unloading.For example, after performing true triaxial unloading tests, Li et al. 3 concluded that the aspect ratio and intermediate principal stress of samples had an impact on the failure mode, peak strength and deformation degree of hard rock.Tomio Horibe 4 concluded that under the condition of a certain loading contact area, the ultimate compressive strength of the material under different loads decreases with the increase of its perimeter (shape effect).Kong et al. 5 carried out the true triaxial compression test on volcanic rocks and found that the brittle characteristics of rock failure are enhanced with the increase of intermediate principal stress.Moomivan, H. and Vutukuri, V.S. studied the effects of size and aspect ratio on the compressive strength of coal and rock samples 6 .Moomivan, H. studied the effect of size on the compressive strength of coal 7 .Moomivan, www.nature.com/scientificreports/H. and Vutukuri, V.S. studied the effect of geometric shape on the compressive strength of columns 8 .These results have provided valuable insight into rock failure characteristics from the laboratory tests' perspectives.
In addition, the rheological properties of underground coal and rock pillars are extremely important for long-term stability and stratum control.Researchers all over the world have conducted extensive rheological test research on rock mechanical properties.For over a decade, Danesh, Fujii, Ladanyi, Ito et al. [9][10][11][12] dedicated themselves to creep test research on the creep characteristics of coal and rock, as well as their influences on permeability and circumferential strain behavior of brittle rock.Chinese scholars Zhao and Yang et al. 13,14 conducted creep tests and analysis on coal and rock under the influence of rock joints.Xia et al. 15 developed and applied the full shear-seepage coupling of rock joints test system.Fu 16 carried out experimental research on the size effect of uniaxial compressive strength of rocks with different height-diameter ratios.Xiao et al. 17 carried out research on anisotropic creep characteristics of quartz-mica schist through triaxial compression creep tests.Meng et al. 18 carried out experimental research on the influences of size effect and strain rate on rock mechanical properties.Under different circumstances, the stability of coal and rock will be affected by the surrounding geological conditions.Huang et al. 19 researched on the creep damage mechanism of coal and rock and the stability of coal pillar on the gob side of the working face.However, the stability differences and failure characteristics of rock samples with different shapes and sizes are the problems that are rarely taken into consideration at present.
Particularly, based on the concept of reducing subsidence, controlling loss, and realizing green mining, researchers have proposed the strip Wongawilli (SW) mining method 20 .This mining method leaves coal pillars with different shapes to control rock strata movement and surface subsidence.Guo et al. [20][21][22] studied the design parameters of underground mining/excavation layout, working face layout, elastic-plastic zone of coal pillar, and stress state in the SW mining method through plentiful surveys and research, and drew useful conclusions.Particularly, the characteristics of reserved coal pillars under room-and-pillar mining are different from those under the SW mining method (Fig. 1).The patterns of coal pillars include rectangular coal pillars, strip coal pillars, rhombic coal pillars, narrow coal pillars, etc.
In this paper, the failure characteristics and rheological properties of samples with different shapes were investigated by means of experimental research, theoretical analysis and numerical simulation.Studying the failure forms, failure processes, and rheological characteristics of samples with different shapes can reveal the essence and laws of the failure of coal pillars with different shapes, providing scientific basis for the design and construction of mining engineering.

Sample preparation
According to the characteristics of reserved coal pillars under SW mining, the size of the single unit of the SW coal pillar, namely the irregular-shaped coal pillar, was scaled by an equal ratio of 1: 200.On this basis, the area of the loading surface of the rock sample was determined, and the sizes of the other five samples with different shapes were determined.Some collected coarse yellow sandstone blocks were processed by a RCD-250 drilling machine, a RLS-100 stone sawing machine, a RG-200 stone grinding machine and a line cutting machine in the laboratory into the following samples respectively: cylindrical samples with a diameter of 100 mm and a height of 100 mm, hexagonal prismatic samples with a side length of 55 mm and an angle of 120°, cubic samples with a side length of 89 mm and an angle of 90°, rhombic samples with a side length of 95.5 mm and an angle of 120°, triangular prismatic samples with a side length of 135 mm and an angle of 60°, and samples with the irregular shape (swallow shape).The loading surface areas of the samples with different shapes were all 0.0079 m 2 .The rock samples obtained by the above-mentioned machines are shown in Fig. 2. The physical and mechanical parameters of rock samples obtained under standard sample size are listed in Table 1.
Conventional uniaxial compression and uniaxial rheological tests were conducted on coarse yellow sandstone samples with different shapes under the same loading surface area with the aid of an RLW-2000 rock triaxial rheometer and a DS2-16B acoustic emission tester.

Test system
Uniaxial mechanical loading failure tests were carried out on samples with different shapes (cylinder, regular hexagon, square, rhombus, equilateral triangle and irregular Swallow shape) by the RLW-2000 rock triaxial rheometer (Fig. 3).The system, which is composed of a fully rigid mechanical loading system and a servo control and data acquisition system, can support a variety of stress path tests such as uniaxial tests, conventional triaxial tests, true triaxial tests, creep tests and cyclic loading and unloading tests.

Acoustic emission testing system
The acoustic emission monitoring system used during the experiment was from the DS2-16B eight channel acoustic emission signal processing equipment and eight AE Amplifier

Uniaxial instantaneous compression failure experimental scheme
Each group of samples were loaded under an initial loading force of 30 KN by a loading rate of 158 N/s until they failed.The deformation degrees of the samples were collected by a digital dial gauge, and the bearing loads were directly collected and controlled by computer software.

Failure characteristics of samples with different shapes under uniaxial instantaneous compression
The morphologies and characteristics of samples with different shapes under the same loading contact area are shown in Fig. 5, from which the following points can be concluded.
1.When rock samples were loaded, the failure of the main shear surface mostly started from one end face and ended at the other end face, connecting both ends.2. Some materials were crushed into powder at the stress concentration position during uniaxial compression instability failure by the testing machine.3. The round sample also underwent tensile failure during loading failure, which was caused by cone shear failure at the end.A relatively complete cone could be taken out from the damaged sample.4. The prismatic samples with sharp corners such as hexagonal, rhomboid and triangle were more likely to be destructed from the tip under the pressure of external load, and the smaller the angle, the greater the deformation degree.5.Under the action of pressure, the sample with the SW special shape (swallow shape) often started to fail at the middle part of the swallow wing and the small diamond rather than at the sharp corner.

Experimental analysis of samples with different shapes under uniaxial compression
According to uniaxial compression test results of samples with different shapes, their peak uniaxial compressive strength decreases with the increase of free surface (gob-side face).Meanwhile, for samples with different shapes, the uniaxial compressive strength ranges from 74.44 MPa under the free surface perimeter of 314 mm (round sample) to 52.49MPa under 660.6 mm (swallow-shaped sample), as shown in Table 3.
As can be seen from the stress-strain curves in Fig. 6, the deformation of rock samples with different shapes can all be divided into four stages, i.e., the compaction stage, the elastic deformation stage, the nonlinear  deformation stage and the strain softening stage.However, these stages present varying characteristics for different samples.Besides, the strain softening stage gradually lengthens with the increase of perimeter, that is, the larger the free surface, the larger the strain value.
The results suggest that the material is obviously brittle, and the stress drops rapidly after the load reaches the peak strength.The stress-strain curves of rock samples with different shapes correspond to different characteristics.Specifically, the stress-strain curve of the round sample approximates that of a conventional standard sample.Those of the square and swallow-shape samples are relatively incomplete, and these samples fail soon after the peak strength is reached, leaving a quite short strain softening stage.The rest three samples do not experience a sharp drop of stress when the peak strength is reached, but show plastic deformation.That is, the round, regular hexagonal, rhombic and triangular samples exhibit a certain brittle-ductility characteristics.

Analysis of acoustic emission data from loading experiments on samples with different shapes
From the acoustic emission monitoring graph, it can be seen that during the loading process of the round specimen, the acoustic emission impact number shows a shape similar to the stress-strain curve, with several states, namely the stage of rapid increase in impact number, the stage of yield reduction, and the final stage of stable  www.nature.com/scientificreports/failure of the specimen.After being loaded for a certain period, the impact number of the rhombus specimen continues to increase, indicating that the specimen's accumulated energy and elastic energy are large.After a period of failure, the swallow-shaped specimen still has a certain load-bearing capacity in the middle, and then fails and becomes unstable again under the action of force.From the acoustic emission monitoring data of the triangular specimen loading, it was found that the failure of the specimen is a continuous process, from the initiation of the crack to the final instability.The failure of the specimen is a continuous steady-state process, indicating that the triangle specimen is the least capable of bearing capacity under force loading.Besides, In the loading of regular rock samples such as round, hexagons, squares, and triangle, the monitoring activity patterns of each free surface acoustic emission channel are nearly similar.Based on the ringing counts and impact times of acoustic emission monitoring in each stage, different shapes can be loaded under the same load area.The data is different, but the difference is not significant, indicating that the shape effect (free surface effect) of regular rock samples is not significant; At the triangular loading acoustic emission monitoring station, it was found that the ringing count and number of impacts were relatively severe, indicating that the triangular sample is prone to damage; Especially in the acoustic emission monitoring of irregular (swallow shaped) specimen loading, it was found that the range values of each channel were very large.It is believed that the stress activity of each free surface of the swallow-shaped specimen under load is intense in the middle region of the swallow-shaped wing.
The acoustic emission test results are shown in Fig. 7.

Selection of numerical simulation model and parameters
In the hope of better and more fully analyzing the shape (free surface) effect of the rock samples, Tyson polygons were constructed with the aid of third-party software Neper to simulate rock internal joints, and then the established rock models with different shapes were imported into 3DEC to simulate the instability failure properties of these samples.3DEC is the abbreviation for 3 Dimension Distinct Element Code, which is the three-dimensional discrete element method program.As the name suggests, 3DEC is a computational analysis program based on the discrete element method as the basic theory to describe the mechanical behavior of discrete media.The Lagrangian solution mode determines that 3DEC has strong universality analysis capabilities in the field of continuous medium mechanics.At the same time, the core idea of the discrete element method endows 3DEC with essential advantages in dealing with non-continuous medium links, especially suitable for the analysis of static and dynamic www.nature.com/scientificreports/response problems of discrete medium under loads (force load, fluid, temperature, etc.), such as the study of medium motion, large deformation, or failure behavior and failure process.3DEC has a wide range of analysis modules in the field of geotechnical engineering, including dynamic analysis, rheological analysis, temperature analysis, joint network flow analysis, and other modules.
Tyson polygon is a subdivision of a spatial plane, characterized by the closest distance from any point within the polygon to its sample points (such as residential areas), the farthest distance from adjacent polygon samples, and each polygon containing only one sample point.Due to the bisection characteristics of Tyson polygons in space, they can be used to solve discretization problems.Neper is a software package for multi crystal generation and meshing.Polycrystals can be 2D or 3D.Neper is built around four modules: module T generates polycrystals as inlays; M module grid polycrystals are described as embedded files; Module S collaborates with FEPX to generate a simulation directory; Module V generates subdivision, mesh, and simulation results of PNG images or VTK files with publication quality (for interactive visualization).
The built model is shown in Fig. 8.The selected values of numerical simulation parameters are given in Table 2.
The solving process of Normal stiffness (Kn) and Shear stiffness (Ks) is as follows : (1) where E is Young's modulus; v is Poisson's ratio; K and G are the bulk and shear moduli, respectively; Kn and Ks, are the normal stiffness and shear stiffness, respectively; factor is a multiplication factor(usually set to 10); △z min is the smallest width of an adjoining zone in the normal direction,0.005m.

Numerical simulation results and analysis
As given in Table 3, the uniaxial compressive strength of samples with different shapes obtained from the numerical simulation experiment ranges from 75.31 MPa of the round sample to 52.38 MPa of the swallow-shaped sample.The peak uniaxial compressive strengths of samples also show a rule of decreasing with the increase of free surface.The comparison of uniaxial compressive strength of different shapes obtained by different experimental methods is shown in Fig. 9.
The peak strengths of samples with different shapes were obtained by 3DEC numerical simulation.The peak strength shows a downward trend overall, but that of the sample with an irregular shape (swallow shape) increases instead.Moreover, under the same loading contact area, the bearing capacity of this sample is larger than that of the triangular one, which provides reliable data for the irregular-shaped coal pillars.
The stress-strain curves of samples with different shapes obtained by 3DEC numerical simulation software are illustrated in Fig. 10.
The simulated uniaxial compression failure characteristics obtained based on the numerical simulation experiment of rock mechanics are presented in Fig. 11.
The following aspects can be seen in the simulated failure experiment on the samples with different shapes.
1.For the regular-shaped samples, the stress is usually concentrated at the center of the sample under the action of pressure, so that the failure develops from the middle.2. The angular samples often start to fail from the edges of the prism, mainly due to the tip effect.3. The round samples has the strongest bearing capacity, followed by the hexagonal, square and rhombic samples which have similar bearing capacities.It is noteworthy that the bearing capacities of the triangular and swallow-shaped samples are quite weak, of which that of the swallow-shaped sample is relatively strong.

Experimental study on rheological instability properties of samples with different shapes
In order to obtain the rheological properties of rock samples with different shapes, rheological tests were performed on samples with six shapes mentioned above.

Rheological test scheme
The test scheme is as follows.
After the pre-test and calculation, the rheological test was performed based on the force-loading control strategy, and the loading rate of the testing machine was set as 158 N/s. 2. Initial load.
Considering the high strength of sandstone, the initial axial load was set as 30 kN. 3. Loading path.
To study the rheological properties of samples in the high-stress area, loads that equaled 70%, 80%, 90% and 100% of their own ultimate strengths were applied in the first, second, third and fourth stages in turn until the samples failed finally.To study the steady-state behavior of specimens with different shapes, each stage of loading was maintained for 24 h.

Rheological test results and analysis
The following findings can be yielded through the stress-strain curves of rock samples with different shapes subject to multi-stage loading (Fig. 12).
1. Axial and tensile splitting failure of rock samples under uniaxial compression is not directly related to the reduction of their bearing capacities.2. The rock samples with different shapes deform to varying degrees finally, and the deformation degree decreases with the increase of free surface area.
Experimental study on the rheological failure characteristics of samples with different shapes Figure 13 exhibits the morphologies and characteristics of rheological failure of samples with different shapes under the same loading contact area.As can be observed from Fig. 13, when rock samples were loaded, the failure of the main shear surface mostly started from one end face and ended at the other end face, connecting both ends.Some materials were crushed into powder at the stress concentration position during uniaxial compression instability failure by the testing machine.The round sample also underwent tensile failure during loading failure, which was caused by the cone shear failure at the end.A relatively complete cone could be taken out from the damaged sample.The prismatic samples with sharp corners such as hexagonal, rhomboid and triangle were more likely to be destructed from the tip under the pressure of external load, and the smaller the angle, the greater the deformation degree.Under the action of pressure, the sample with the SW special shape (swallow shape) often started to fail at the middle part of the swallow wing and the small diamond rather than at the sharp corner.In short, with respect to failure mode, the coarse yellow sandstone samples with different shapes underwent brittle failure under the influence of pressure.In terms of failure mechanism, the round, hexagonal and square samples mainly showed compressive shear failure, while the rhombic, triangular and swallow-shaped samples presented tensile fracture or fracturing failure.4).It can be found from Table 4 that the stability of the rock samples weakens with the increase of free surface.Based on the mathematical theory, the bearing efficiency formula of free surface of coarse yellow sandstone samples with different shapes under the same loading surface is obtained: where x is the ratio of loading surface perimeter; y is the ratio of steady-state rheological time to final failure time.
As can be seen from the fitting curve in Fig. 14, when the free surface area (loading surface perimeter) accounts for over 20% of that of the round sample, the ratio of steady-state rheological time to final failure time no longer falls significantly.That is to say, the steady-state rheological time becomes similar regardless of the increase of free surface area after the ratio of free surface area exceeds 20%, which verifies the rationality of using swallow-shaped coal pillars in the SW mining method.This study guided the design of coal pillars in the continuous mining face of Wangtaipu Coal Mine and conducted a reliability analysis for the stability of the coal pillars left in the coal mine face.www.nature.com/scientificreports/According to the uniaxial compression and uniaxial rheological compression test results of rocks with different shapes, the changes of peak uniaxial compressive strength under the two conditions are displayed in Fig. 15.Clearly, the peak uniaxial rheological strength of the round sample is far lower than its uniaxial compressive strength, while the peak uniaxial rheological strengths of the samples with the other shapes are close to their uniaxial compressive strengths.

Figure 1 .
Figure 1.Shapes of coal pillars reserved under different mining methods.
40 dB acoustic emission sensors produced by Beijing Soft Island Times Company.(Acoustic emission frequency range: 50-400 kHz, center frequency: 150 kHz).The eight-channel acoustic emission signal interface of the DS2-16B acoustic emission detector system is placed on the different sides of the tested sample rock mass in the rock mechanics testing machine.The characteristic parameters of the acoustic emission signal loaded on the sample are shown in Fig. 4.

Figure 2 .
Figure 2. Finished products of columnar rock samples with different shapes.

Figure 3 .
Figure 3. Rheological test system for acoustic emission and electromagnetic radiation signals from rock samples.

Figure 4 .
Figure 4. Characteristic parameter diagram of loading AE signal of coal pillar sample.

Figure 5 .
Figure 5. Failure characteristics of pillar-like rock samples with different shapes under uniaxial compression.

Figure 7 .
Figure 7. Acoustic emission monitoring data chart for loading different shaped specimens.

Figure 8 .Table 2 .
Figure 8. Construction of Tyson polygon models with different shapes.

Figure 9 .
Figure 9.Comparison of peak strengths of pillar-like rock samples with different shapes obtained from 3DEC numerical simulation and physical test.

Figure 10 .
Figure 10.Stress strain curves of pillar-like rock samples with different shapes in 3DEC numerical simulation software.

Figure 11 . 4 .
Figure 11.3DEC numerical simulation results and experimental results of plastic failure zones of pillar-like rock samples with different shapes.

Figure 12 .
Figure 12.Stress-strain curves of pillar-like rock samples with different shapes subject to multi-stage loading by the RLW-2000 triaxial rheometer.

2 ×Figure 13 .
Figure 13.Rheological failure modes of pillar-like rock samples with different shapes.

Figure 14 .
Figure 14.Relation curve between ratio of steady-state rheological time to final failure time and free surface area of pillar-like rock samples with different shapes.

Figure 15 .
Figure 15.Comparison of peak strengths of pillar-like rock samples with different shapes under uniaxial compression and rheological conditions.

Table 3 .
Uniaxial compressive strengths of samples with different shapes obtained from different experiments.

Table 4 .
Statistics of steady-state rheological time and failure time of samples with different shapes.t 1 in the table represents the time when there was no failure during specimen loading; T 2 represents the total loading time of the specimen in the case of final instability.The period between t 1 and t 2 is the time from the specimen's micro failure to the final instability.